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Modular latticesModular lattices
A modular lattice is a lattice L = (L,∨,∧) that satisfies the modular identity: (( x∧z) ∨y) ∧z = ( x∧z) ∨( y∧z) .
A modular lattice is a lattice L = (L,∨,∧) that satisfies the modular law: x ≤ z ⇒ ( x∨y) ∧z ≤ x∨( y∧z) .
A modular lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic to the pentagon N5 =
Let L and M be modular lattices. A morphism from L to M is a function h : L→M that is a homomorphism: h(x∨y) = h(x)∨h(y) and h(x∧y) = h(x)∧h(y).
M3 = is the smallest nondistributive modular lattice. By a result of [Richard Dedekind, Über die von drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900) 371--403] this lattice occurs as a sublattice of every nondistributive modular lattice.
| Classtype | variety |
| Equational theory | undecidable
[Ralph Freese,
Free modular lattices,
Trans. Amer. Math. Soc.
261
(1980)
81--91
MRreview]
[Christian Herrmann, On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983) 513--527 MRreview] |
| Quasiequational theory | undecidable [L. Lipshitz, The undecidability of the word problems for projective geometries and modular lattices, Trans. Amer. Math. Soc. 193 (1974) 171--180 MRreview] |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | no |
| Definable principal congruences | no |
| Equationally definable principal congruences | no |
| Amalgamation property | no |
| Strong amalgamation property | no |
| Epimorphisms are surjective | no |